First DS. Recommendations? (With Giveaway)

Fire Emblem: Awakening

Pokemon X/Y, Super Mario 3D World, Fire Emblem: Awakening and Zelda: A Link Between Worlds are good games.

Getting "The Legend Of Zelda: Ocarina Of Time 3D" is kind of a no-brainer. I've almost finished "A Link Between Worlds" too now and it's quite good. Could be better, especially LONGER, but it's still a decent game.

Getting Pokémon X or Y might be an idea too.

For RPGs? Since Fire Emblem has already been mentioned, look into the Etrian Odyssey and Shin Megami Tensei games available for 3DS and DS. I suppose you could also try Pokemon X/Y, or Tales of the Abyss if you've never played the PS2 version.

Fire Emblem: Awakening, yep.

Try getting Super Robot Wars Original Generation Endless Frontier. Nostalgia is also a good choice. I wish I could recommend Golden Sun Dark Dawn, but I haven't played it. You could also get the three DS Castlevania games. (Mind that all of these are not 3DS games, they are for the original DS.)

Professor Layton games are great, if you like to think

Luminous Arc for srpg fix

Inazuma Eleven for football with rpg elements fix

Castlevania Order of Eclessia for side scrolling action rpg fix

... i just like rpgs :P

If you are in for a hunt, Monster Hunter 3.

Radiant Historia

Try Scribblenauts :)

Life With Horses 3D

Ace Attorney, Dragon Quest

999, Ghost Trick, Hotel Dusk or any Ace Attorney, Castlevania or Megami Tensei game, also I've heard Radiant Historia is pretty great.

Super Scribblenauts and the Pokemon games are pretty good choices.

Ace Attorney, Super Scribblenauts, Kingdom Hearts 385/2 Days (!!) and Fire Emblem are some of the more well-known good games, but if you're willing to look around for some lesser known awesome games:
Digimon World Championship: pretty good and overlooked title, I wasted tons of hours playing it
Monster Rancher DS: Think Tamagotchi with battles
Geometry Wars: Retro Evolved
Chrono Trigger: Classic RPG that's still well worth it's price
Monster Racers: A surprisingly fun platform-race take on the whole monster-capturing concept

If you're looking to buy some eShop stuff: Sujin Taisen Number battles (or Number Battle in America) is one of the best games I have ever played in my life! (and I've played quite alot) Pop Island Paperfields is a very nice Download play title for 2 Euros, and Steamworld Dig is basically Motherload (the flash game) with a progression system.

The Legend Of Zelda: A Link Between Worlds

No one's said animal crossing yet...

Pokemon and Animal Crossing :D

Those are the reason I'm buying the same DS. xP

Dead or Alive: Dimensions. Also Legend of Zelda: Ocarina of Time if you didn't play it before on N64.

Legend of Zelda series
Star Fox
Pokémon X and Y
Super Smash Bros. (Not sure if the new one is out already...if not get it when you can :p)
Monster Hunter
Fire Emblem

Obviously Pokemon, but what about Disgaea? It's not a 3DS game (I think it's from the regular DS, but it should be compatible), but I had lots of fun with the PS2 version.

Pokemon Y&X or OoT

How is it possible that Dragon Quest IX hasn't been mentioned yet?It's one of the best RPGs for DS,maybe even the best.

Professor Layton games. They are damned fun puzzles.